Complex Analogues of the Rolle's Theorem
2000 Mathematics Subject Classification: 30C10. Classical Rolle’s theorem and its analogues for complex algebraic polynomials are discussed. A complex Rolle’s theorem is conjectured.
Extention of Apolarity and Grace Theorem
MSC 2010: 30C10 The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of several well-known...
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